AB and CD are two equal and parallel chords of the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2)) =1`. Tangents to the ellipse at A and B intersect at P and tangents at C and D at Q. The line PQ

Options

A. passes through the origin
B. is bisected at the origin
C. cannot pass through the origin
D. is not bisected at the origin

Answer

A, B

Explanation

Let P and Q be the points `(alpha, beta)` and `(alpha_(1),beta_(1))`
`rArr` Equations of Ab and CD are `(x)/(a) alpha + (y)/(b) beta =1` and `(x)/(a) alpha_(1) +(y)/(b) beta_(1) =1` (Chord of contact)
These lines are parallel
`rArr (alpha)/(alpha_(1)) = (beta)/(beta_(1)) =k`
Also `(alpha^(2))/(alpha^(2)) +(beta^(2))/(b^(2)) = (alpha_(1)^(2))/(a^(2)) + (beta_(1)^(2))/(b^(2))`
`rArr (alpha)/(alpha_(1)) = (beta)/(beta_(1)) =-1`
`rArr PQ` passes through origin and is bisected at the origin.